Beystadiums or simply Stadiums are products released by Takara Tomy and Hasbro to facilitate Beyblade battles. Hasbro also refers to their products as Arenas.
Metal Saga
Takara Tomy
Metal Saga Beystadiums released by Takara Tomy initially were stadiums with surfaces that gave advantages to certain Beyblades (ex. Attack Type Beystadium). Specialty Beystadiums such as Ultimate Bey Ta Stadium were also released.
With the introduction of the Synchrome System, Takara Tomy introduced a new design meant to facilitate the "Zero-G" gimmick.
Hasbro
Hasbro's Metal Fusion Beystadiums initially featured the same design, with two penalty pockets. The exception to this was the Super Vortex Beystadium. This trend continued into Metal Masters, until Metal Fury introduced more varied Beystadium designs. Hasbro initially re-used molds for their Shogun Steel releases, before releasing their own version of the Zero-G Beystadium, the Salamander Cyclone Beystadium.
Burst Series
While Takara Tomy's designs seem to follow the same basic design cues as previous systems, Hasbro introduced new Beystadiums to coincide with the release of their exclusive compatible systems. The SlingShock Beystadium for example, has "rails" that allow SlingShock System Beyblades to ride. The HyperSphere Beystadium has a "vertical wall", "brink", and "battle ring" that allows Hypersphere System Beyblades to drop down from.
X Series
Stadiums released in the X toyline have gears that help activate the Xtreme Dash gimmick.